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Properties

Label 336.7.bh.c
Level $336$
Weight $7$
Character orbit 336.bh
Analytic conductor $77.298$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,7,Mod(145,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.145");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 336.bh (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(77.2981720963\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 33x^{6} + 2x^{5} + 701x^{4} - 28x^{3} + 6468x^{2} + 5488x + 38416 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (9 \beta_{2} - 18) q^{3} + (20 \beta_{2} - 5 \beta_1 + 15) q^{5} + ( - 5 \beta_{7} + 5 \beta_{6} + \cdots + 113) q^{7}+ \cdots + ( - 243 \beta_{2} + 243) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (9 \beta_{2} - 18) q^{3} + (20 \beta_{2} - 5 \beta_1 + 15) q^{5} + ( - 5 \beta_{7} + 5 \beta_{6} + \cdots + 113) q^{7}+ \cdots + (4131 \beta_{7} + 972 \beta_{6} + \cdots - 124659) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 108 q^{3} + 210 q^{5} + 608 q^{7} + 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 108 q^{3} + 210 q^{5} + 608 q^{7} + 972 q^{9} - 2058 q^{11} - 3780 q^{15} - 11244 q^{17} - 21834 q^{19} - 4482 q^{21} - 15504 q^{23} - 6550 q^{25} + 35316 q^{29} + 51060 q^{31} + 55566 q^{33} - 71460 q^{35} + 20282 q^{37} + 101682 q^{39} - 387812 q^{43} + 51030 q^{45} + 55212 q^{47} - 277780 q^{49} + 101196 q^{51} - 336174 q^{53} + 393012 q^{57} + 560454 q^{59} + 850728 q^{61} - 26730 q^{63} + 826380 q^{65} + 947882 q^{67} - 147192 q^{71} - 533034 q^{73} + 176850 q^{75} - 1848102 q^{77} + 6260 q^{79} - 236196 q^{81} + 560040 q^{85} - 476766 q^{87} + 413460 q^{89} - 256074 q^{91} - 459540 q^{93} + 170880 q^{95} - 1000188 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 33x^{6} + 2x^{5} + 701x^{4} - 28x^{3} + 6468x^{2} + 5488x + 38416 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 125 \nu^{7} + 10474 \nu^{6} - 188235 \nu^{5} + 1867654 \nu^{4} - 7794823 \nu^{3} + \cdots + 265757772 ) / 18120004 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 687 \nu^{7} + 4419 \nu^{6} - 23077 \nu^{5} + 72651 \nu^{4} - 337317 \nu^{3} + 2423085 \nu^{2} + \cdots + 19535908 ) / 18120004 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1356 \nu^{7} - 108791 \nu^{6} + 866022 \nu^{5} - 8426099 \nu^{4} + 15546218 \nu^{3} + \cdots - 1433259800 ) / 18120004 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4022 \nu^{7} - 116327 \nu^{6} + 473948 \nu^{5} - 3427703 \nu^{4} + 1884320 \nu^{3} + \cdots - 276695356 ) / 18120004 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3609 \nu^{7} + 12501 \nu^{6} - 65283 \nu^{5} + 754377 \nu^{4} - 954243 \nu^{3} + 6854715 \nu^{2} + \cdots + 55265532 ) / 4530001 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17229 \nu^{7} + 50292 \nu^{6} - 724881 \nu^{5} + 350472 \nu^{4} - 11327325 \nu^{3} + \cdots - 67584720 ) / 9060002 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37215 \nu^{7} - 457372 \nu^{6} + 2090607 \nu^{5} - 12626944 \nu^{4} + 27670891 \nu^{3} + \cdots - 1069683916 ) / 18120004 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 2\beta_{7} - \beta_{5} - 8\beta_{4} + 2\beta_{3} + 38\beta_{2} + 8\beta _1 + 8 ) / 84 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 8\beta_{7} - 3\beta_{6} + 4\beta_{5} - 4\beta_{4} - 4\beta_{3} + 652\beta_{2} - 2\beta _1 - 650 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 46\beta_{7} + 11\beta_{6} - 57\beta_{5} - 128\beta_{4} - 92\beta_{3} + 128\beta_{2} - 256\beta _1 - 2122 ) / 84 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -67\beta_{7} - 131\beta_{5} + 37\beta_{4} - 67\beta_{3} - 6040\beta_{2} - 37\beta _1 - 37 ) / 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 836 \beta_{7} - 481 \beta_{6} - 418 \beta_{5} + 1552 \beta_{4} + 418 \beta_{3} - 21542 \beta_{2} + \cdots + 20766 ) / 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3712 \beta_{7} - 1703 \beta_{6} + 5415 \beta_{5} + 2290 \beta_{4} + 7424 \beta_{3} - 2290 \beta_{2} + \cdots + 259592 ) / 42 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 34694\beta_{7} + 116227\beta_{5} - 47440\beta_{4} + 34694\beta_{3} + 1768546\beta_{2} + 47440\beta _1 + 47440 ) / 84 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1 - \beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
1.77014 + 3.06597i
2.56933 + 4.45021i
−1.36222 − 2.35944i
−1.97725 − 3.42469i
1.77014 − 3.06597i
2.56933 − 4.45021i
−1.36222 + 2.35944i
−1.97725 + 3.42469i
0 −13.5000 + 7.79423i 0 −65.9934 − 38.1013i 0 334.046 + 77.8600i 0 121.500 − 210.444i 0
145.2 0 −13.5000 + 7.79423i 0 −52.3293 − 30.2124i 0 17.4716 − 342.555i 0 121.500 − 210.444i 0
145.3 0 −13.5000 + 7.79423i 0 41.1897 + 23.7809i 0 138.771 + 313.674i 0 121.500 − 210.444i 0
145.4 0 −13.5000 + 7.79423i 0 182.133 + 105.155i 0 −186.289 − 288.003i 0 121.500 − 210.444i 0
241.1 0 −13.5000 − 7.79423i 0 −65.9934 + 38.1013i 0 334.046 − 77.8600i 0 121.500 + 210.444i 0
241.2 0 −13.5000 − 7.79423i 0 −52.3293 + 30.2124i 0 17.4716 + 342.555i 0 121.500 + 210.444i 0
241.3 0 −13.5000 − 7.79423i 0 41.1897 − 23.7809i 0 138.771 − 313.674i 0 121.500 + 210.444i 0
241.4 0 −13.5000 − 7.79423i 0 182.133 − 105.155i 0 −186.289 + 288.003i 0 121.500 + 210.444i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.7.bh.c 8
4.b odd 2 1 42.7.g.b ✓ 8
7.d odd 6 1 inner 336.7.bh.c 8
12.b even 2 1 126.7.n.b 8
28.d even 2 1 294.7.g.b 8
28.f even 6 1 42.7.g.b ✓ 8
28.f even 6 1 294.7.c.a 8
28.g odd 6 1 294.7.c.a 8
28.g odd 6 1 294.7.g.b 8
84.j odd 6 1 126.7.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.7.g.b ✓ 8 4.b odd 2 1
42.7.g.b ✓ 8 28.f even 6 1
126.7.n.b 8 12.b even 2 1
126.7.n.b 8 84.j odd 6 1
294.7.c.a 8 28.f even 6 1
294.7.c.a 8 28.g odd 6 1
294.7.g.b 8 28.d even 2 1
294.7.g.b 8 28.g odd 6 1
336.7.bh.c 8 1.a even 1 1 trivial
336.7.bh.c 8 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 210 T_{5}^{7} - 5925 T_{5}^{6} + 4331250 T_{5}^{5} + 357598125 T_{5}^{4} + \cdots + 21\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 27 T + 243)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 92\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 61\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 62\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 31\!\cdots\!28)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 42\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 85\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 55\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 56\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 29\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 74\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
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